| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Simultaneous equations with logarithms |
| Difficulty | Standard +0.8 This requires converting between logarithm bases, manipulating simultaneous equations with logs, and substituting log_x(y) = -3 to find log_2(x). It goes beyond routine log law application to require multi-step algebraic manipulation and insight about base conversion, making it moderately harder than average. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4(x-2) \Rightarrow 4x-8\), \(2x+1 \Rightarrow x \leq 4.5\) | M1A1 | M1: proceeds to \(x \leq \ldots\) after multiplying out; A1: \(x\leq 4.5\) or equivalent e.g. \(\{x: x\leq 4.5\}\), \((-\infty, 4.5]\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Roots are \(1.5, -5\) | B1 | Critical values \(1.5, -5\) |
| Chooses outside: \(x < -5\), \(x > 1.5\) | M1A1 | M1: chooses outside values; A1: accept \(x<-5, x>1.5\) or \(x<-5\) or \(x>1.5\); do not accept '\(x<-5\) and \(x>1.5\)' or '\(-5>x>1.5\)' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x < -5\), \(1.5 < x \leq 4.5\) | B1 | cao; must be two distinct regions with two inequalities; if candidate writes \(x\leq 4.5\), \(x<-5\), \(x>1.5\) it is B0 |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4(x-2) \Rightarrow 4x-8$, $2x+1 \Rightarrow x \leq 4.5$ | M1A1 | M1: proceeds to $x \leq \ldots$ after multiplying out; A1: $x\leq 4.5$ or equivalent e.g. $\{x: x\leq 4.5\}$, $(-\infty, 4.5]$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Roots are $1.5, -5$ | B1 | Critical values $1.5, -5$ |
| Chooses outside: $x < -5$, $x > 1.5$ | M1A1 | M1: chooses outside values; A1: accept $x<-5, x>1.5$ or $x<-5$ or $x>1.5$; do not accept '$x<-5$ and $x>1.5$' or '$-5>x>1.5$' |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x < -5$, $1.5 < x \leq 4.5$ | B1 | cao; must be two distinct regions with two inequalities; if candidate writes $x\leq 4.5$, $x<-5$, $x>1.5$ it is B0 |
---$$\begin{gathered}
2 \log _ { 2 } y = 5 - \log _ { 2 } x \\
\log _ { x } y = - 3
\end{gathered}$$
for $x > 0 , y > 0$\\
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\hfill \mbox{\textit{Edexcel C12 2016 Q2 [6]}}